Introduction

NiTi shape memory alloys (SMAs) can recover large inelastic strains due to their B2 ↔ B19' martensitic transformations [1,2,3], as manifested in the shape memory effect [4,5,6] and the pseudoelastic behaviour [7,8,9]. The deformation and recovery of NiTi is a complex process involving different mechanisms, including elastic deformation, plastic deformation, and crystallographic deformation associated with the stress-induced transformation [10, 11] or martensite reorientation [12,13,14]. Each of these deformation mechanisms has a different Poisson’s ratio. For example, the B2 ↔ B19' transformation is practically an isochoric process [15], thus it has a theoretical Poisson’s ratio of 0.5 whereas for elastic deformation the Poisson’s ratio is typically 3.0 ~ 3.4. In this regard, a NiTi component may exhibit different Poisson’s ratios in different stages during a deformation process or in different parts experiencing different deformation mechanisms [16,17,18]. Such behaviour may be of concern for the design of NiTi-based materials, particularly those in intimate mix or contact with other materials and in miniature scales, for example NiTi nanocomposites [19,20,21], multilayer laminates [22, 23], NiTi thin films on substrates [16], and MEMS using these films [24, 25].

The Poisson’s ratio of NiTi has been measured by various means. Mita et al. measured the acoustic response of a Ti-50.9 at.% Ni alloy during tensile deformation via stress-induced martensitic transformation and reported an average Poisson’s ratio of 0.46 [26]. Namazu et al. used in situ X-ray diffraction technique to measure the ratio of the out-of-plane (lateral) to the in-plane (axial) strains of near-equiatomic NiTi thin films and obtained a Poisson’s ratio of 0.31 for elastic deformation [24]. Qui et al. used in situ neutron diffraction method and determined the Poisson’s ratio of the B19' martensite during tensile and compressive deformation to be ~ 0.39 [27]. Obviously, these Poisson’s ratios determined using diffraction techniques are for pure elastic lattice deformation and do not reflect the behaviour of the transformation/reorientation deformation or plastic deformation. Bewerse et al. reported Poisson’s ratio values of 0.3–0.45 based on digital image correlation measurement for perforated NiTi plates deforming via stress-induced martensitic transformation [28]. Such a wide range of variation is indicative of the different deformation mechanisms dominant in different parts of the perforated plate. The Poisson’s ratios of NiTi single crystals have also been investigated by means of ab initio calculations [29,30,31,32,33,34]. The Poisson’s ratio of elastic lattice deformation of NiTi was found to vary between 0.31 and 0.59 depending on the crystal structure and crystallographic orientation. It is to be noted that Poisson’s ratios of the values larger than 0.5 are possible for anisotropic materials, e.g. single crystals.

NiTi components often deform via stress-induced martensitic transformation in a Lüders-type manner under tension [8, 35, 36]. During this process, the axial strains inside and outside the Lüders band are different in both the magnitude and the deformation mechanism. This imposes a special mechanical complexity when NiTi is used in close contact with other materials, such as in intermetallic composites [37, 38], thin films or coatings on NiTi substrates [39, 40], and in MEMS devices [24, 41], where elastic deformation, plastic deformation, and crystallographic deformation may all mingle and constrain within one system. In this regard, it is of interest to characterise and to understand the Poisson’s behaviour of NiTi. In this study, the variation of apparent Poisson’s ratio of pseudoelastic NiTi alloy during different stages of tensile deformation was analysed by means of digital image correlation (DIC) technique [42] and analytical modelling to provide this critical data for understanding the material behaviour.

Materials and Methods

A Ti-50.9 at.% Ni sheet alloy of 0.53 mm in thickness was used in this study. Dog-bone shaped samples of 0.53 × 2.5 × 30 mm in gauge section dimension for tensile testing were cut from the plate by means of electrical discharge machining. A stochastic pattern was created on the surface of each sample using flat white and black paints to enable in situ DIC measurement during tensile testing. Tensile testing was conducted using a KQL (WDT-10 model) device under displacement-control condition with a slow strain rate of ~ 1 × 10–4/s to maintain a near isothermal condition. The testing temperature was 301 K. In situ DIC measurements were performed to characterise the local strain evolution during tensile deformation.

Experimental Analysis

Figure 1 shows the DIC measurement of a tensile pseudoelastic deformation cycle of a NiTi sample. Figure 1a shows the true stress—true strain curve of the sample. The true strain was obtained from DIC measurement. Figure 1b shows the DIC images of the local axial strain \({\varepsilon }_{x}\) distribution in the sample at different moments of the Lüders band evolution. The numerals labelled on each of the images correspond to those marked on the stress–strain curve. Two virtual miniature extensometers, marked \({L}_{x}\) and \({L}_{y}\), are applied at point P on the sample to measure the axial strain \({\varepsilon }_{x}\) along the \(x\) direction and the lateral strain \({\varepsilon }_{y}\) in the \(y\) direction, respectively. The nominal gauge length of the virtual extensometers is 1.5 mm. Figure 1c shows the evolutions of \({\varepsilon }_{x}\) and \({\varepsilon }_{y}\) during the process of the deformation.

Fig. 1
figure 1

DIC characterisation of the tensile deformation behaviour of a pseudoelastic Ti-50.9 at.% Ni alloy. a True stress—true strain curve of the sample. b DIC images of axial strain \({\varepsilon }_{x}\) of the sample at different moments during the pseudoelastic deformation cycle. c Evolutions of the axial and lateral strains at point P during the pseudoelastic deformation cycle as acquired by DIC

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Analytical Modelling

This section establishes the analytical relationships for the apparent Poisson’s ratio of NiTi at different stages of the pseudoelastic deformation cycle. The deformation process, as presented in Fig. 1a, is considered in this analysis to contain only the elastic deformation and the deformation associated with the stress-induced B2 ↔ B19' martensitic transformation. Parameters \({\nu }_{e}\), \({\nu }_{t}\) and \({\nu }_{a}\) represent the elastic, transformation and apparent Poisson’s ratios of the sample, respectively. These Poisson’s ratios can be generically written as:

$${\nu }_{i}=-\frac{{\varepsilon }_{{y}_{i}}}{{\varepsilon }_{{x}_{i}}} \;\;(i=e, t, a)$$
(1)

The apparent strain consisting of elastic and transformation contributions can be written as:

$${\varepsilon }_{{i}_{a}}={\varepsilon }_{{i}_{e}}+{\varepsilon }_{{i}_{t}} \;\; (i=x, y)$$
(2)

Figure 2a shows definitions of the parameters in different stages of the Lüders-type deformation used for analytical modelling. For this analysis, the loading path of the sample is divided into 4 stages, including the initial apparent linear “elastic” stage O-①, the nonlinear stage ①-②, the transformation stage ②-③ from the nucleation of the Lüders band until the band front reaches point P, and the transformation stage ③-④ from the moment when the Lüders band reaches point P until the end of the stress plateau. The numerals are the same as those defined in Fig. 1. Here, \({E}_{e}\) is the true elastic modulus of the B2 austenite, \({E}_{a}\) is the apparent modulus of the austenite determined from the stress–strain curve during the initial loading up to point ①, and \({\sigma }_{t}\) is the transformation stress of the B2 → B19' transformation.

Fig. 2
figure 2

Different stages of tensile deformation behaviour of Ti-50.9 at.% Ni alloy. a Definitions of the various parameters in different stages of the stress–strain curve used for analytical modelling. b Variation of the apparent Poisson’s ratio during loading. c Variation of the apparent Poisson’s ratio during unloading

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Stage I: \(0\le \sigma <{\sigma }_{1}\)

In this stage, the apparent stress–strain variation is linear, but the strain contains contribution from the martensitic transformation in addition to elasticity. It is also noted that the transformation in this stage is uniform, i.e. blended spatially with the austenite matrix undistinguishable to the DIC strain field measurement. Therefore, the stress–strain relations for the true elastic and the apparent linear behaviours of the material can be written respectively as:

$${\varepsilon }_{{x}_{e}}=\frac{\sigma }{{E}_{e}}$$
(3)
$${\varepsilon }_{{x}_{a}}=\frac{\sigma }{{E}_{a}}$$
(4)

Using Eqs. (1) and (2), the apparent Poisson’s ratio can be written as:

$${\nu }_{a}=-\frac{{\varepsilon }_{{y}_{e}}+{\varepsilon }_{{y}_{t}}}{{\varepsilon }_{{x}_{e}}+{\varepsilon }_{{x}_{t}}}=\frac{{{\nu }_{e}\varepsilon }_{{x}_{e}}+{{\nu }_{t}\varepsilon }_{{x}_{t}}}{{\varepsilon }_{{x}_{a}}}=\frac{{{\nu }_{e}\varepsilon }_{{x}_{e}}+{{\nu }_{t}(\varepsilon }_{{x}_{a}}-{\varepsilon }_{{x}_{e}})}{{\varepsilon }_{{x}_{a}}}$$
(5)

Using Eqs. (3), (4) and (5), the apparent Poisson’s ratio for stage I is obtained as:

$${\nu }_{a}={\nu }_{t}-\frac{{E}_{a}}{{E}_{e}}({\nu }_{t}-{\nu }_{e})$$
(6)

It is clear that the apparent Poisson’s ratio of the alloy in this stage is determined by the Poisson’s ratios of elasticity and the phase transformation, and the ratio of \({E}_{a}/{E}_{e}\), which expresses the partition or the relative contributions of the two strains to the total deformation.

Stage II: \({\sigma }_{1}\le \sigma <{\sigma }_{t}\)

In this stage, the stress–strain curve is fitted with a nonlinear function as:

$$\sigma ={a}_{0}+{a}_{1}{\varepsilon }_{{x}_{a}}+{a}_{2}{\varepsilon }_{{x}_{a}}^{2}$$
(7)

Using Eqs. (3), (5) and (7), the apparent Poisson’s ratio for stage II is obtained as:

$$\nu _{a} \, = \,\nu _{t} \, - \,\frac{1}{{E_{e} }}\,(a_{0} \,\varepsilon _{{x_{a} }}^{{ - 1}} \, + \,a_{1} \, + \,a_{2} \,\varepsilon _{{x_{a} }} )\,(\nu _{t} \, - \,\nu _{e} )$$
(8)

It is clear in this expression that the apparent Poisson’s ratio in this stage is determined by the Poisson’s ratios of elasticity and the stress-induced phase transformation, the elastic modulus of the austenite, and the apparent strain. This implies that the apparent Poisson’s ratio in this stage varies with the progression of the deformation (apparent strain) and is not a single value. This is obviously due to the progressively increased non-elastic contribution to the deformation (as evidenced by the curved shape of the stress–strain curve).

Stage III: \({\varepsilon }_{{a}_{2}}\le \varepsilon \le {\varepsilon }_{{a}_{3}}\)

This stage starts at the moment when the Lüders band is nucleated, i.e. the beginning of the stress plateau, and ends at the moment when the transformation band front reaches point P, i.e. when the apparent strain is \({\varepsilon }_{{a}_{3}}\) as denoted in Fig. 2a. During this stage, point P remains outside the Lüders band and its condition remains unchanged in a state corresponding to point ②. Using Eqs. (1) and (2), the apparent Poisson’s ratio can be expressed as:

$${\nu }_{a}=-\frac{{\varepsilon }_{{y}_{e}}+{\varepsilon }_{{y}_{t}}}{{\varepsilon }_{{x}_{e}}+{\varepsilon }_{{x}_{t}}}=\frac{{{\nu }_{e}\varepsilon }_{{e}_{2}}+{{\nu }_{t}\varepsilon }_{{x}_{t}}}{{\varepsilon }_{{a}_{2}}}=\frac{{{\nu }_{e}\varepsilon }_{{e}_{2}}+{{\nu }_{t}(\varepsilon }_{{a}_{2}}-{\varepsilon }_{{e}_{2}})}{{\varepsilon }_{{a}_{2}}}$$
(9)

where \({\varepsilon }_{{e}_{2}}\) and \({\varepsilon }_{{a}_{2}}\) are the elastic and apparent strains respectively at point ②, as denoted in Fig. 2a. Using Eqs. (3) and (9), the apparent Poisson’s ratio at point P for stage III is obtained as:

$${\nu }_{a}={\nu }_{t}-\frac{{\sigma }_{t}}{{E}_{e}{\varepsilon }_{{a}_{2}}}({\nu }_{t}-{\nu }_{e})$$
(10)

This gives a particular value for the apparent Poisson’s ratio given that \({\sigma }_{t}\) and \({\varepsilon }_{{a}_{2}}\) are fixed values which can be determined from the stress–strain curve.

Stage IV: \({\varepsilon }_{{a}_{3}}\le \varepsilon \le {\varepsilon }_{{a}_{4}}\)

This stage starts at the moment when the Lüders band reaches point P, i.e. when the apparent strain is \({\varepsilon }_{{a}_{3}}\), and ends at the end of the stress plateau of the B2 → B19' transformation. During this stage, point P remains inside the Lüders band and its condition remains unchanged in a state corresponding to point ③. Using Eqs. (1) and (2), the apparent Poisson’s ratio can be written as:

$${\nu }_{a}=-\frac{{\varepsilon }_{{y}_{e}}+{\varepsilon }_{{y}_{t}}}{{\varepsilon }_{{x}_{e}}+{\varepsilon }_{{x}_{t}}}=\frac{{{\nu }_{e}\varepsilon }_{{e}_{2}}+{{\nu }_{t}\varepsilon }_{{x}_{t}}}{{\varepsilon }_{{a}_{4}}}=\frac{{{\nu }_{e}\varepsilon }_{{e}_{2}}+{{\nu }_{t}(\varepsilon }_{{a}_{4}}-{\varepsilon }_{{e}_{2}})}{{\varepsilon }_{{a}_{4}}}$$
(11)

where \({\varepsilon }_{{a}_{4}}\) is the apparent strain at the end of the stress plateau as denoted in Fig. 2a. Using Eqs. (3) and (11), the apparent Poisson’s ratio at point P for stage IV is obtained as:

$${\nu }_{a}={\nu }_{t}-\frac{{\sigma }_{t}}{{E}_{e}{\varepsilon }_{{a}_{4}}}({\nu }_{t}-{\nu }_{e})$$
(12)

It is to be noted that stages III and IV are both for the Lüders band deformation process over the stress plateau. The reason why two different Poisson’s ratio values are calculated in Eqs. (10) and (12) is that during this process the sample has two distinctive sections, one inside the Lüders band and one outside, which deform by different mechanisms.

With the establishment of Eqs. (6), (8), (10) and (12), variation of the Poisson’s ratio during the process of deformation can be modelled. For this, the parameters used in these equations are determined from the true stress–strain curve presented in Fig. 1a, as presented below. For the parameters related to the nonlinear sections of the stress–strain curve, i.e. in sections ①-② and ⑤-⑥, curve fitting functions were used.

$${E}_{a}=64.11 GPa, {a}_{0}=-303.64 MPa, {a}_{1}=162730 MPa, {a}_{2}=-8875200 MPa, {\sigma }_{t}=420 MPa, {\varepsilon }_{{a}_{2}}=0.01, {\varepsilon }_{{a}_{4}}=0.063$$
(13)

The elastic and transformation properties of NiTi were determined based on information from the literature, as below [15, 27, 29, 32]:

$${E}_{e}=100 GPa, {\nu }_{e}=0.35, {\nu }_{t}=0.5$$
(14)

Figure 2b shows the variation of the apparent Poisson’s ratio (\({\nu }_{a}\)) of the sample at point P during the loading path of the pseudoelastic deformation cycle. The red curve represents the experimentally measured apparent Poisson’s ratio based on the \({\varepsilon }_{x}\) and \({\varepsilon }_{y}\) data presented in Fig. 1c. The black curve represents the result calculated using Eqs. (6), (8), (10), (12) and the data presented in Eqs. (13) and (14). The red dot on the Y axis indicates the Poisson’s ratio \({\nu }_{e}=0.35\) for elasticity of the B2 austenite of NiTi [27, 29].

It is seen that upon loading, \({\nu }_{a}\) remained unchanged in stage I at 0.39. This value is higher than \({\nu }_{e}=0.35\). This implies the participation of the stress-induced martensitic transformation in this stage. This is consistent with the observation of the slightly increased temperature of the sample (due to the release of the latent heat) during this stage of deformation [7, 43] and the measurement of the low apparent elastic modulus values [1, 11, 44,45,46]. The apparent Poisson’s ratio gradually increased during stage II (①-②), indicating the increased relative contribution of the stress-induced transformation prior to the onset of the Lüders-type deformation. In stage III (②-③), when the observation point remained outside of the Lüders band, the Poisson’s ratio remained practically constant at 0.44. Advancing into stage IV (③-④), when the Lüders band front reached point P, \({\nu }_{a}\) jumped to 0.48 and then remained constant until the end of stage IV. This is obviously due to the dominant contribution of the stress-induced B2 → B19' martensitic transformation, which has a Poisson’s ratio of 0.50. It is also seen that the analytical solution matches well the experimental result.

The above analytical relationships developed for the loading path can also be applied to the unloading path by substituting points ①, ②, ③ and ④ with points ⑤, ⑥, ⑦ and ⑧ in the analysis. For this, the elastic properties of martensite and the B19' → B2 transformation properties determined from the true stress–strain curve are used, as the following:

$${E}_{a}=31.45 GPa, {a}_{0}=4530.4 MPa, {a}_{1}=-165570 MPa, {a}_{2}=1575800 MPa, {\sigma }_{t}=170 MPa, {\varepsilon }_{{a}_{6}}=0.052, {\varepsilon }_{{a}_{8}}=0.004, {E}_{e}=50 GPa, {\nu }_{e}=0.35, {\nu }_{t}=0.5$$
(15)

Using Eqs. (6), (8), (10) and (12) and the data presented in Eq. (15) and by taking point ④ as the reference point for calculating stress and strain, the variation of the apparent Poisson’s ratio over the unloading path can be computed, which is presented as the green curve in Fig. 2c. The blue curve represents the experimentally measured apparent Poisson’s ratio at point P. As observed, \({\nu }_{a}\) remained relatively constant at 0.40 within stage ④-⑤. Upon further unloading, it gradually increased from 0.4 at point ⑤ to 0.45 at point ⑥, which is the beginning of the reverse stress plateau. It remained almost unchanged during the reverse B19' → B2 transformation until the reverse transformation front reached point P at the global strain of ~ 0.008 (section ⑥-⑦). Then, \({\nu }_{a}\) increased from 0.45 to 0.48 within ⑦-⑧ as the reverse transformation passed point P. It is also noted that in both Fig. 2b and c, the experimental curve deviated from the predicted trend with significantly higher or lower \({\nu }_{a}\) values at points ③ and ⑦. This is apparently related to the passing of the transformation front, within which the local strain field is highly inhomogeneous. These values apparently cannot be taken as the true value of Poisson’s ratio of the sample.

Conclusion

  1. (1)

    The experimental measurement and analytical modelling demonstrate that the apparent Poisson’s ratio of NiTi changes during the pseudoelastic tensile deformation cycle, increasing from 0.40 within the initial apparent linear elastic deformation range to 0.44 at the onset of the Lüders-type deformation and further to 0.48 within the Lüders band.

  2. (2)

    The initial value of 0.40 is above that expected for elasticity of the B2 phase of NiTi. This implies the participation of non-elastic deformation during the apparent linear stage of deformation. This is attributed to the stress-induced B2 → B19' transformation. The increase of the Poisson’s ratio to 0.44 and the apparent curved stress–strain variation further demonstrate the occurrence of the transformation prior to the onset of the Lüders-type deformation.

  3. (3)

    The increase of the Poisson’s ratio to 0.48 is attributed to the increased contribution of the stress-induced B2 → B19' transformation

  4. (4)

    The proposed analytical model predicts well the variation of the apparent Poisson’s ratio of NiTi alloy during pseudoelastic deformation.